Heat flow studies play an important role in understanding the state of the lithosphere. For modeling the conductive heat transfer in the crust the important controlling parameters are the radiogenic heat production and the thermal conductivity. In case of a stabilized continental crust the conductive heat transfer in steady state condition is a reasonably good approximation. Several authors have studied the evolution of the basal heat flow from a deterministic point of view. However due to the heterogeneous nature of the crust researchers are now solving the heat conduction problem from a stochastic point of view.
Quantification of perturbations in the temperatures and heat flow using stochastic analytical and random simulation techniques have been carried out by several authors. The uncertainty in the heat flow using a least squares inversion technique incorporating uncertainties in the temperature and thermal conductivities has been done, Tectonophysics, Vol 121, 1985 by Vausser et al. The effect of variation in heat source on the surface heat flow has also been studied, Journal Geophysical Research, V 91, 1986, by Vasseur and Singh, Geophysical Research Letters, V14, 1987, by Nielsen.
The small perturbation method to solve the stochastic heat conduction equation has been used to solve the 1-D steady state equation with uncertainties in the heat sources to obtain the mean temperature field along with its error bounds , Geophysical Journal International, 135, 1998, by Srivastava and Singh. Several researchers have been using the numerical method, the random simulation method, to model the error structure in the thermal field incorporating uncertainties in the controlling thermal parameters, Tectonophysics, V156, 1988 by Royer and Danis, Marine and Petroleum Geology, V 14, 1997, by Gallagher et al, Tectonophysics, V 306, 1999a,b, by Jolinen and Kukkonen.
The diffusion problems in stochastic framework are now being solved by yet another approach called the decomposition method, Journal of Hydrology, V 169, 1995, by Serrano. In this invention the same approach as given in Geophysical Journal International, V 138, 1999, by Srivastava and Singh, has been used to solve the stochastic heat conduction equation incorporating Gaussian uncertainties in the thermal conductivity to obtain the solution to the mean and variance in the subsurface heat flow field.